Proving Limits

a) Show that if C is any straight line through (0,0) then [tex]\lim_{(x,y) \to (0,0)}[/tex] along C exists and equals 1.

b) Show that the limit as (x,y) -> 0 doesn't exist.

2. Relevant equations

3. The attempt at a solution

I really need help with this question! I know that x2 is a parabola and any straight line through the origin either intersects the parabola at some point and remains above it until 0 is reached (or lies on y=0, in which case [tex]y \leq 0[/tex] and [tex]f(x,y) = 1[/tex]) but still I don't know how to "prove" part a).

I know that x2 is a parabola and any straight line through the origin either intersects the parabola at some point and remains above it until 0 is reached (or lies on y=0, in which case [tex]y \leq 0[/tex] and [tex]f(x,y) = 1[/tex]) but still I don't know how to "prove" part a).

But you've done it!

If, sufficiently close to the origin, it remains above the parabola until 0 is reached, then the value along that part of the line is 1, so the limit exists …